The triangle $ABC$ is inscribed in a circle. Two ants crawl out of points $B$ and $C$ at the same time. They crawl along the arc $BC$ towards each other so that the product of the distances from them to point $A$ remains unchanged. Prove that during their movement (until the moment of meeting), the straight line passing through the ants touches some fixed circle.

We are given a lattice and two pebbles $A$ and $B$ that are placed at two lattice points. At each step we are allowed to relocate one of the pebbles to another lattice point with the condition that the distance between pebbles is preserved. Is it possible after finite number of steps to switch positions of the pebbles?

Suppose $\{a_n\}_{n=1}^\infty$ is a sequence. The partial sums $\{s_n\}_{n=1}^\infty$ are defined by $$s_n=\sum_{i=1}^na_i.$$ The Cesàro sums are then defined as $\{A_n\}_{n=1}^\infty$, where $$A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i.$$ Let $a_n=(-1)^{n+1}$. What is the limit of the Cesàro sums of $\{a_n\}_{n=1}^\infty$ as $n$ goes to infinity?

We call a pair $(a,b)$ of positive integers, $a<391$, [i]pupusa[/i] if $$\textup{lcm}(a,b)>\textup{lcm}(a,391)$$ Find the minimum value of $b$ across all [i]pupusa[/i] pairs. Fun Fact: OMCC 2017 was held in El Salvador. [i]Pupusa[/i] is their national dish. It is a corn tortilla filled with cheese, meat, etc.

After years at sail, you and your crew have found the island that houses the great treasure of Scottybeard, the greatest pirate to ever sail the high seas. The island takes the shape of a unit square, and the treasure (which we treat as a single point) could be buried under any point on the island. To assist you in finding his treasure, Scotty has left a peculiar instrument. To use this instrument, you may draw any directed line (possibly one that never hits the island!), and the instrument will tell you whether the treasure lies to the "left" or the "right" of the line.* [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((1,0.5)--(0.8,0.42),arrow=Arrow()); draw((0.8,0.42)--(0.6,0.34),arrow=Arrow()); draw((0.6,0.34)--(0.4,0.26),arrow=Arrow()); draw((0.4,0.26)--(0.2,0.18),arrow=Arrow()); draw((0.2,0.18)--(0,0.1)); label("``Right''", (0.5,0.55)); label("``Left''", (0.8,0.2)); [/asy] However, Scotty also left a trap! If the instrument ever reports ``left'' three times in a row or ``right'' three times in a row, the island will suddenly sink into the sea, submerging the treasure forever and drowning you and your crew! You want to avoid this at all costs. To minimize the amount of energy spent digging, you would like to narrow down the set of possible locations of the treasure to be as small as possible. However, Scotty left one last trick; you can only use the instrument 12 times before it breaks! Devise an algorithm to use the instrument no more than 12 times that can never result in the island sinking and narrows the worst-case space of possible locations of the treasure to have as small an area as possible. * [size=75]Where "left" or "right" is taken with respect to an observer walking along the line in its designated direction. There is also a probability zero chance the treasure is precisely on the line; this won't affect anything, but for the sake of clarity let's say the instrument reports "left" in this case.[/size] [b]Scoring:[/b] An algorithm that achieves a worst-case area of $K$ will be awarded: [list] [*] 1 point for any $K<1$ [*] 10 points for $K=\tfrac 14$ [*] 20 points for $\tfrac 1{128}<K<\tfrac 14$ [*] 30 points for $K=\tfrac 1{128}$ [*] 50 points for $K_{\text{min}}<K<\tfrac 1 {128}$ [*] 75 points for $K=K_{\text{min}}$ [*] 100 points for $K=K_{\text{min}}$, with a proof that this is optimal [/list] (where $K_{\text{min}}$ is the smallest possible worst-case area, which we are not disclosing to avoid giving anything away) [i]Proposed by Connor Gordon[/i]

Point $P$ lies inside triangle of sides of length $3, 4, 5$. Show that if distances between $P$ and vertices of triangle are rational numbers then distances from $P$ to sides of triangle are rational numbers too.

Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.

Let $ABC$ be an acute triangle with orthocenter $H$. The circumcircle of the triangle $BHC$ intersects $AC$ a second time in point $P$ and $AB$ a second time in point $Q$. Prove that $H$ is the circumcenter of the triangle $APQ$. [i](Karl Czakler)[/i]

On the semicircle with diameter $AB$ and center $O$ point $D$ is marked. Points $E$ and $F$ are the midpoints of minor arcs $AD$ and $BD$ respectively. It turned out that the line connecting orthocenters of $ADF$ and $BDE$ passes through $O$ Find $\angle AOD$

Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

A real number $x$ is chosen uniformly at random from the interval $(0,10).$ Compute the probability that $\sqrt{x}, \sqrt{x+7},$ and $\sqrt{10-x}$ are the side lengths of a non-degenerate triangle.

Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.

Let $f(x) = \sqrt{4x^2 - 4x^4}$. Let $A$ be the number of real numbers $x$ that satisfy $$f(f(f(\dots f(x)\dots ))) = x,$$ where the function $f$ is applied to $x$ 2020 times. Compute $A \pmod {1000}$. [i]Proposed by Timothy Qian[/i]

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

Let $S$ be the sum of all products $ab$ where $a$ and $b$ are distinct elements of the set $\{1,2,...,46\}$. Prove that $47$ divides $S$.

In triangle $ABC$, $AB>AC.$ The bisector of $\angle BAC$ meets $BC$ at $D.$ $P$ is on line $DA,$ such that $A$ lies between $P$ and $D$. $PQ$ is tangent to $\odot(ABD)$ at $Q.$ $PR$ is tangent to $\odot(ACD)$ at $R.$ $CQ$ meets $BR$ at $K.$ The line parallel to $BC$ and passing through $K$ meets $QD,AD,RD$ at $E,L,F,$ respectively. Prove that $EL=KF.$